Means Of Walsh-Fourier Series Research Articles

On the weighted maximal operators of Marcinkiewicz type Cesàro means of two-dimensional Walsh-Fourier series

In this paper we investigate the behaviour of the weighted maximal operators of Marcinkiewicz type (C,?)-means ??,* p (f) := supn?P |??n(f)|/ n2/p?(2+?) in the Hardy space Hp(G2) (0 < ? < 1 and p < 2/(2 + ?)). It is showed that the maximal operators ??,* p (f) are bounded from the dyadic Hardy space Hp(G2) to the Lebesgue space Lp(G2), and that this is in a sense sharp. It was also proved a strong convergence theorem for the Marcinkiewicz type (C, ?) means of Walsh-Fourier series in Hp(G2).

Pointwise convergence of cone-like restricted two-dimensional Fejér means of Walsh-Fourier series

For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β−1 ≤ n1/n2 ≤ β is provided with some fixed parameter β ≥ 1. In this paper we generalize the result of Gat and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property.

On the Rate of Convergence of Cesàro Means of Walsh-Fourier Series

The aim of this paper is to make a complete investigation concerning the interaction between the rate of convergence of Cesàro means of Walsh-Fourier series and the modulus of continuity. We give the best possible sufficient conditions with respect to the modulus of continuity that implies the convergence at a given rate. We also give the best necessary conditions. These questions are studied in L p (1 ≤ p < ∞) and in uniform norms. As a consequence, we receive the best results for the Lipschitz classes. The solution of a problem of F. Móricz and A. H. Siddiqi (1992, J. Approx. Theory 70, 375-389), i.e., the characterization of the Favard (saturation) classes of the Cesàro summation, can be derived from our theorems.

Approximation by Nörlund means of Walsh-Fourier series

We study the rate of approximation by Nörlund means for Walsh-Fourier series of a function in L p and, in particular, in Lip(α, p) over the unit interval [0, 1), where α > 0 and 1 ⩽ p ⩽ ∞. In case p = ∞, by L p we mean C W , the collection of the uniformly W-continuous functions over [0, 1). As special cases, we obtain the earlier results by Yano, Jastrebova, and Skvorcov on the rate of approximation by Cesàro means. Our basic observation is that the Nörlund kernel is quasi-positive, under fairly general assumptions. This is a consequence of a Sidon type inequality. At the end, we raise two problems.

Certain estimates of approximation of functions by Ces�ro means of Walsh-Fourier series

Certain estimates of approximation of functions by Ces�ro means of Walsh-Fourier series